scholarly journals Clustering of rapidly settling, low-inertia particle pairs in isotropic turbulence. Part 1. Drift and diffusion flux closures

2019 ◽  
Vol 871 ◽  
pp. 450-476 ◽  
Author(s):  
Sarma L. Rani ◽  
Vijay K. Gupta ◽  
Donald L. Koch
Keyword(s):  
Velocity Gradient ◽  
Dissipation Rate ◽  
Isotropic Turbulence ◽  
Diffusion Flux ◽  
Fluid Velocity ◽  
Polar Angle ◽  
Turbulent Dissipation ◽  
Drift Flux ◽  
Particle Pairs ◽  
And Diffusion

In this two-part study, we present the development and analysis of a stochastic theory for characterizing the relative positions of monodisperse, low-inertia particle pairs that are settling rapidly in homogeneous isotropic turbulence. In the limits of small Stokes number and Froude number such that $Fr\ll St_{\unicode[STIX]{x1D702}}\ll 1$, closures are developed for the drift and diffusion fluxes in the probability density function (p.d.f.) equation for the pair relative positions. The theory focuses on the relative motion of particle pairs in the dissipation regime of turbulence, i.e. for pair separations smaller than the Kolmogorov length scale. In this regime, the theory approximates the fluid velocity field in a reference frame following the primary particle as locally linear. In this part 1 paper, we present the derivation of closure approximations for the drift and diffusion fluxes in the p.d.f. equation for pair relative positions $\boldsymbol{r}$. The drift flux contains the time integral of the third and fourth moments of the ‘seen’ fluid velocity gradients along the trajectories of primary particles. These moments may be analytically resolved by making approximations regarding the ‘seen’ velocity gradient. Accordingly, two closure forms are derived specifically for the drift flux. The first invokes the assumption that the fluid velocity gradient along particle trajectories has a Gaussian distribution. In the second drift closure, we account for the correlation time scales of dissipation rate and enstrophy by decomposing the velocity gradient into the strain-rate and rotation-rate tensors scaled by the turbulent dissipation rate and enstrophy, respectively. An analytical solution to the p.d.f. $\langle P\rangle (r,\unicode[STIX]{x1D703})$ is then derived, where $\unicode[STIX]{x1D703}$ is the spherical polar angle. It is seen that the p.d.f. has a power-law dependence on separation $r$ of the form $\langle P\rangle (r,\unicode[STIX]{x1D703})\sim r^{\unicode[STIX]{x1D6FD}}$ with $\unicode[STIX]{x1D6FD}\sim St_{\unicode[STIX]{x1D702}}^{2}$ and $\unicode[STIX]{x1D6FD}<0$, analogous to that for the radial distribution function of non-settling pairs. An explicit expression is derived for $\unicode[STIX]{x1D6FD}$ in terms of the drift and diffusion closures. The $\langle P\rangle (r,\unicode[STIX]{x1D703})$ solution also shows that, for a given $r$, the clustering of $St_{\unicode[STIX]{x1D702}}\ll 1$ particles is only weakly anisotropic, which is in conformity with prior observations from direct numerical simulations of isotropic turbulence containing settling particles.

scholarly journals Clustering of rapidly settling, low-inertia particle pairs in isotropic turbulence. Part 2. Comparison of theory and DNS

2019 ◽  
Vol 871 ◽  
pp. 477-488 ◽  
Author(s):  
Sarma L. Rani ◽  
Rohit Dhariwal ◽  
Donald L. Koch
Keyword(s):  
Reasonable Agreement ◽  
Isotropic Turbulence ◽  
Diffusion Flux ◽  
Stochastic Theory ◽  
Polar Angle ◽  
Harmonic Coefficient ◽  
Gravitational Acceleration ◽  
Particle Clustering ◽  
Particle Pairs ◽  
And Diffusion

Part 1 (Rani et al. J. Fluid Mech., vol. 871, 2019, pp. 450–476) of this study presented a stochastic theory for the clustering of monodisperse, rapidly settling, low-Stokes-number particle pairs in homogeneous isotropic turbulence. The theory involved the development of closure approximations for the drift and diffusion fluxes in the probability density function (p.d.f.) equation for the pair relative positions $\boldsymbol{r}$. In this part 2 paper, the theory is quantitatively analysed by comparing its predictions of particle clustering with data from direct numerical simulations (DNS) of isotropic turbulence containing particles settling under gravity. The simulations were performed at a Taylor micro-scale Reynolds number $Re_{\unicode[STIX]{x1D706}}=77.76$ for three Froude numbers $Fr=\infty ,0.052,0.006$, where $Fr$ is the ratio of the Kolmogorov scale of acceleration and the magnitude of gravitational acceleration. Thus, $Fr=\infty$ corresponds to zero gravity, and $Fr=0.006$ to the highest magnitude of gravity among the three DNS cases. For each $Fr$, particles of Stokes numbers in the range $0.01\leqslant St_{\unicode[STIX]{x1D702}}\leqslant 0.2$ were tracked in the DNS, and particle clustering quantified both as a function of separation and the spherical polar angle. We compared the DNS and theory values for the exponent $\unicode[STIX]{x1D6FD}$ characterizing the power-law dependence of clustering on separation. The $\unicode[STIX]{x1D6FD}$ from the $Fr=0.006$ DNS case are in reasonable agreement with the theoretical predictions obtained using the second drift closure (referred to as DF2). To quantify the anisotropy in clustering, we calculated the leading–order coefficient in the spherical harmonics expansion of the p.d.f. of pair relative positions. The coefficients predicted by the theory (DF2) again show reasonable agreement with those calculated from the DNS clustering data for $Fr=0.006$. However, we note that in spite of the high magnitude of gravity, the clustering is only marginally anisotropic both in DNS and theory. The theory predicts that the spherical harmonic coefficient scales with $\unicode[STIX]{x1D6FD}(=\unicode[STIX]{x1D6FD}_{2}St_{\unicode[STIX]{x1D702}}^{2})$, where $\unicode[STIX]{x1D6FD}_{2}$ is the ratio of the drift and diffusion flux coefficients. Since the drift flux, and thereby $\unicode[STIX]{x1D6FD}_{2}$, is seen to decrease with gravity for $St_{\unicode[STIX]{x1D702}}<1$, the anisotropy is also correspondingly diminished.

A stochastic model for the relative motion of high Stokes number particles in isotropic turbulence

2014 ◽  
Vol 756 ◽  
pp. 870-902 ◽  
Author(s):  
Sarma L. Rani ◽  
Rohit Dhariwal ◽  
Donald L. Koch
Keyword(s):  
Relative Motion ◽  
Isotropic Turbulence ◽  
Diffusion Tensor ◽  
Fluid Velocity ◽  
Planck Equation ◽  
Stokes Number ◽  
Diffusion Current ◽  
Particle Pairs ◽  
Kolmogorov Scale ◽  
Fokker Planck

AbstractThe probability density function (PDF) kinetic equation describing the relative motion of inertial particle pairs in a turbulent flow requires closure of the phase-space diffusion current. A novel analytical closure for the diffusion current is presented that is applicable to high-inertia particle pairs with Stokes numbers $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}{\mathit{St}}_r \gg 1$. Here ${\mathit{St}}_r$ is a Stokes number based on the time scale $\tau _r$ of eddies whose size scales with pair separation $r$. In the asymptotic limit of ${\mathit{St}}_r \gg 1$, the pair PDF kinetic equation reduces to an equation of Fokker–Planck form. The diffusion tensor characterizing the diffusion current in the Fokker–Planck equation is equal to $1/\tau _v^2$ multiplied by the time integral of the Lagrangian correlation of fluid relative velocities along particle-pair trajectories. Here, $\tau _v$ is the particle viscous relaxation time. Closure of the diffusion tensor is achieved by converting the Lagrangian correlations of fluid relative velocities ‘seen’ by pairs into Eulerian fluid-velocity correlations at pair separations that remain essentially constant during time scales of $O(\tau _r)$; the pair centre of mass, however, is not stationary and responds to eddies with time scales comparable to or smaller than $\tau _v$. For isotropic turbulence, Eulerian fluid-velocity correlations may be expressed as Fourier transforms of the velocity spectrum tensor, enabling us to derive a closed-form expression for the diffusion tensor. A salient feature of this closure is that it has a single, unique form for pair separations spanning the entire spectrum of turbulence scales, unlike previous closures that involve velocity structure functions with different forms for the integral, inertial subrange, and Kolmogorov-scale separations. Using this closure, Langevin equations, which are statistically equivalent to the Fokker–Planck equation, were solved to evolve particle-pair relative velocities and separations in stationary isotropic turbulence. The Langevin equation approach enables the simulation of the full PDF of pair relative motion, instead of only the first few moments of the PDF as is the case in a moments-based approach. Accordingly, PDFs $\varOmega (U|r)$ and $\varOmega (U_r|r)$ are computed and presented for various separations $r$, where the former is the PDF of relative velocity $U$ and the latter is the PDF of the radial component of relative velocity $U_r$, both conditioned upon the separation $r$. Consistent with the direct numerical simulation (DNS) study of Sundaram & Collins (J. Fluid Mech., vol. 335, 1997, pp. 75–109), the Langevin simulations capture the transition of $\varOmega (U|r)$ from being Gaussian at integral-scale separations to an exponential PDF at Kolmogorov-scale separations. The radial distribution functions (RDFs) computed from these simulations also show reasonable quantitative agreement with those from the DNS study of Février, Simonin & Legendre (Proceedings of the Fourth International Conference on Multiphase Flow, New Orleans, 2001).

Dissipation rate estimation from PIV in zero-mean isotropic turbulence

2008 ◽  
Vol 46 (3) ◽  
pp. 499-515 ◽  
Author(s):  
J. de Jong ◽  
L. Cao ◽  
S. H. Woodward ◽  
J. P. L. C. Salazar ◽  
L. R. Collins ◽  
...  
Keyword(s):  
Dissipation Rate ◽  
Isotropic Turbulence ◽  
Rate Estimation

Scale-dependent alignment, tumbling and stretching of slender rods in isotropic turbulence

2018 ◽  
Vol 860 ◽  
pp. 465-486 ◽  
Author(s):  
Nimish Pujara ◽  
Greg A. Voth ◽  
Evan A. Variano
Keyword(s):  
Isotropic Turbulence ◽  
Fluid Velocity ◽  
Inertial Range ◽  
Small Scale ◽  
Strain Rate Tensor ◽  
Scaling Exponents ◽  
Slender Body Theory ◽  
Velocity Statistics ◽  
Rigid Rods ◽  
Dissipation Range

We examine the dynamics of slender, rigid rods in direct numerical simulation of isotropic turbulence. The focus is on the statistics of three quantities and how they vary as rod length increases from the dissipation range to the inertial range. These quantities are (i) the steady-state rod alignment with respect to the perceived velocity gradients in the surrounding flow, (ii) the rate of rod reorientation (tumbling) and (iii) the rate at which the rod end points move apart (stretching). Under the approximations of slender-body theory, the rod inertia is neglected and rods are modelled as passive particles in the flow that do not affect the fluid velocity field. We find that the average rod alignment changes qualitatively as rod length increases from the dissipation range to the inertial range. While rods in the dissipation range align most strongly with fluid vorticity, rods in the inertial range align most strongly with the most extensional eigenvector of the perceived strain-rate tensor. For rods in the inertial range, we find that the variance of rod stretching and the variance of rod tumbling both scale as $l^{-4/3}$, where $l$ is the rod length. However, when rod dynamics are compared to two-point fluid velocity statistics (structure functions), we see non-monotonic behaviour in the variance of rod tumbling due to the influence of small-scale fluid motions. Additionally, we find that the skewness of rod stretching does not show scale invariance in the inertial range, in contrast to the skewness of longitudinal fluid velocity increments as predicted by Kolmogorov’s $4/5$ law. Finally, we examine the power-law scaling exponents of higher-order moments of rod tumbling and rod stretching for rods with lengths in the inertial range and find that they show anomalous scaling. We compare these scaling exponents to predictions from Kolmogorov’s refined similarity hypotheses.

scholarly journals A Dynamic Flight Model for Slocum Gliders and Implications for Turbulence Microstructure Measurements

2019 ◽  
Vol 36 (2) ◽  
pp. 281-296 ◽  
Author(s):  
Lucas Merckelbach ◽  
Anja Berger ◽  
Gerd Krahmann ◽  
Marcus Dengler ◽  
Jeffrey R. Carpenter
Keyword(s):  
Dissipation Rate ◽  
Input Parameter ◽  
Weather Conditions ◽  
Water Velocity ◽  
Doppler Velocity ◽  
Electromagnetic Current ◽  
Model Errors ◽  
Turbulent Dissipation ◽  
Adverse Weather ◽  
Flight Model

Abstract The turbulent dissipation rate ε is a key parameter to many oceanographic processes. Recently, gliders have been increasingly used as a carrier for microstructure sensors. Compared to conventional ship-based methods, glider-based microstructure observations allow for long-duration measurements under adverse weather conditions and at lower costs. The incident water velocity U is an input parameter for the calculation of the dissipation rate. Since U cannot be measured using the standard glider sensor setup, the parameter is normally computed from a steady-state glider flight model. As ε scales with U2 or U4, depending on whether it is computed from temperature or shear microstructure, respectively, flight model errors can introduce a significant bias. This study is the first to use measurements of in situ glider flight, obtained with a profiling Doppler velocity log and an electromagnetic current meter, to test and calibrate a flight model, extended to include inertial terms. Compared to a previously suggested flight model, the calibrated model removes a bias of approximately 1 cm s−1 in the incident water velocity, which translates to roughly a factor of 1.2 in estimates of the dissipation rate. The results further indicate that 90% of the estimates of the dissipation rate from the calibrated model are within a factor of 1.1 and 1.2 for measurements derived from microstructure temperature sensors and shear probes, respectively. We further outline the range of applicability of the flight model.

An Investigation of Crossing Trajectory Effects in Turbulent Shear Flow (Keynote)

2005 ◽  
Author(s):  
Lionel Thomas ◽  
Benoiˆt Oesterle´
Keyword(s):  
Shear Flow ◽  
Isotropic Turbulence ◽  
Fluid Velocity ◽  
Stokes Number ◽  
Turbulent Shear ◽  
Inertial Particles ◽  
Turbulent Shear Flow ◽  
Velocity Magnitude ◽  
Dispersed Particles ◽  
Lagrangian Tracking

The dispersion of small inertial particles moving in a homogeneous, hypothetically stationary, shear flow is investigated using both theoretical analysis and numerical simulation, under one-way coupling approximation. In the theoretical approach, the previous studies are extended to the case of homogeneous shear flow with a corresponding anisotropic spectrum. As it is impossible to obtain a closed theoretical solution without some drastic simplifications, the motion of dispersed particles is also investigated using kinematic simulation where random Fourier modes are generated according to a prescribed anisotropic spectrum with a superimposed linear mean fluid velocity profile. The combined effects of particle Stokes number and dimensionless drift velocity (magnitude and direction) are investigated by computing the statistics from Lagrangian tracking of a large number of particles in many flow field realizations, and comparison is made between the observed effects in shear flow and in isotropic turbulence.

scholarly journals Vertical velocity and turbulence aspects during Mistral events as observed by UHF wind profilers

2004 ◽  
Vol 22 (11) ◽  
pp. 3927-3936 ◽  
Author(s):  
J.-L. Caccia ◽  
V. Guénard ◽  
B. Benech ◽  
B. Campistron ◽  
P. Drobinski
Keyword(s):  
Boundary Layer ◽  
Planetary Boundary Layer ◽  
Vertical Velocity ◽  
Convective Boundary Layer ◽  
Dissipation Rate ◽  
Convective Boundary ◽  
Turbulent Dissipation ◽  
The Alps ◽  
Rhone Valley ◽  
Wind Profilers

Abstract. The general purpose of this paper is to experimentally study mesoscale dynamical aspects of the Mistral in the coastal area located at the exit of the Rhône-valley. The Mistral is a northerly low-level flow blowing in southern France along the Rhône-valley axis, located between the French Alps and the Massif Central, towards the Mediterranean Sea. The experimental data are obtained by UHF wind profilers deployed during two major field campaigns, MAP (Mesoscale Alpine Program) in autumn 1999, and ESCOMPTE (Expérience sur Site pour COntraindre les Modèles de Pollution atmosphériques et de Transports d'Emission) in summer 2001. Thanks to the use of the time evolution of the vertical profile of the horizontal wind vector, recent works have shown that the dynamics of the Mistral is highly dependent on the season because of the occurrence of specific synoptic patterns. In addition, during summer, thermal forcing leads to a combination of sea breeze with Mistral and weaker Mistral due to the enhanced friction while, during autumn, absence of convective turbulence leads to substantial acceleration as low-level jets are generated in the stably stratified planetary boundary layer. At the exit of the Rhône valley, the gap flow dynamics dominates, whereas at the lee of the Alps, the dynamics is driven by the relative contribution of "flow around" and "flow over" mechanisms, upstream of the Alps. This paper analyses vertical velocity and turbulence, i.e. turbulent dissipation rate, with data obtained by the same UHF wind profilers during the same Mistral events. In autumn, the motions are found to be globally and significantly subsident, which is coherent for a dry, cold and stable flow approaching the sea, and the turbulence is found to be of pure dynamical origin (wind shears and mountain/lee wave breaking), which is coherent with non-convective situations. In summer, due to the ground heating and to the interactions with thermal circulation, the vertical motions are less pronounced and no longer have systematic subsident charateristics. In addition, those vertical motions are found to be much less developed during the nighttimes because of the stabilization of the nocturnal planetary boundary layer due to a ground cooling. The enhanced turbulent dissipation-rate values found at lower levels during the afternoons of weak Mistral cases are consistent with the installation of the summer convective boundary layer and show that, as expected in weaker Mistral events, the convection is the preponderant factor for the turbulence generation. On the other hand, for stronger cases, such a convective boundary layer installation is perturbed by the Mistral.

Sign in / Sign up

Export Citation Format

Share Document